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Question Number 20914 by Tinkutara last updated on 07/Sep/17
The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has (1) Only purely imaginary roots (2) All real roots (3) Two real and two purely imaginary roots (4) Neither real nor purely imaginary roots
$$\mathrm{The}\:\mathrm{quadratic}\:\mathrm{equation}\:{p}\left({x}\right)\:=\:\mathrm{0}\:\mathrm{with} \\ $$$$\mathrm{real}\:\mathrm{coefficients}\:\mathrm{has}\:\mathrm{purely}\:\mathrm{imaginary} \\ $$$$\mathrm{roots}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{equation}\:{p}\left({p}\left({x}\right)\right)\:=\:\mathrm{0} \\ $$$$\mathrm{has} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Only}\:\mathrm{purely}\:\mathrm{imaginary}\:\mathrm{roots} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{All}\:\mathrm{real}\:\mathrm{roots} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Two}\:\mathrm{real}\:\mathrm{and}\:\mathrm{two}\:\mathrm{purely}\:\mathrm{imaginary} \\ $$$$\mathrm{roots} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{Neither}\:\mathrm{real}\:\mathrm{nor}\:\mathrm{purely}\:\mathrm{imaginary} \\ $$$$\mathrm{roots} \\ $$
Answered by dioph last updated on 08/Sep/17
p(x) = ax^2 +c s≡((±i(√(4ac)))/(2a))=((±i(√(ac)))/a)=±i(√(c/a)) p(p(x)) = a(ax^2 +c)^2 +c = a(a^2 x^4 +2acx^2 +c^2 )+c = a^3 x^4 +2a^2 cx^2 +ac^2 +c u ≡ x^2 p(p(x))=0 ⇔ a^3 u^2 +2a^2 cu+(ac^2 +c)=0 ⇒ u=((−2a^2 c±(√(4a^4 c^2 −4a^4 c^2 −4a^3 c)))/(2a^3 )) ⇒ u = ((−c±i(√(c/a)))/a) = ((−c+s)/a) ⇒ x = (√u) is neither real nor purely imaginary
$${p}\left({x}\right)\:=\:{ax}^{\mathrm{2}} +{c} \\ $$$${s}\equiv\frac{\pm{i}\sqrt{\mathrm{4}{ac}}}{\mathrm{2}{a}}=\frac{\pm{i}\sqrt{{ac}}}{{a}}=\pm{i}\sqrt{\frac{{c}}{{a}}} \\ $$$${p}\left({p}\left({x}\right)\right)\:=\:{a}\left({ax}^{\mathrm{2}} +{c}\right)^{\mathrm{2}} +{c} \\ $$$$=\:{a}\left({a}^{\mathrm{2}} {x}^{\mathrm{4}} +\mathrm{2}{acx}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)+{c} \\ $$$$=\:{a}^{\mathrm{3}} {x}^{\mathrm{4}} +\mathrm{2}{a}^{\mathrm{2}} {cx}^{\mathrm{2}} +{ac}^{\mathrm{2}} +{c} \\ $$$${u}\:\equiv\:{x}^{\mathrm{2}} \\ $$$${p}\left({p}\left({x}\right)\right)=\mathrm{0}\:\Leftrightarrow\:{a}^{\mathrm{3}} {u}^{\mathrm{2}} +\mathrm{2}{a}^{\mathrm{2}} {cu}+\left({ac}^{\mathrm{2}} +{c}\right)=\mathrm{0} \\ $$$$\Rightarrow\:{u}=\frac{−\mathrm{2}{a}^{\mathrm{2}} {c}\pm\sqrt{\mathrm{4}{a}^{\mathrm{4}} {c}^{\mathrm{2}} −\mathrm{4}{a}^{\mathrm{4}} {c}^{\mathrm{2}} −\mathrm{4}{a}^{\mathrm{3}} {c}}}{\mathrm{2}{a}^{\mathrm{3}} } \\ $$$$\Rightarrow\:{u}\:=\:\frac{−{c}\pm{i}\sqrt{{c}/{a}}}{{a}}\:=\:\frac{−{c}+{s}}{{a}} \\ $$$$\Rightarrow\:{x}\:=\:\sqrt{{u}}\:\mathrm{is}\:\mathrm{neither}\:\mathrm{real}\:\mathrm{nor}\:\mathrm{purely} \\ $$$$\mathrm{imaginary} \\ $$
Commented by Tinkutara last updated on 08/Sep/17
Thank you very much Sir!
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

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